Q 6.51 - Intermediate Algebra

Question

Factor: $8x^{2}y-24xy+18y$.

Step-by-Step Solution

Answer

Factors of $8x^{2}y-24xy+18y$ are $2y{\left(2x-3\right)}^2$.

Step 1. Given information.

Given an expression $8x^{2}y-24xy+18y$.

Step 2. Find if GCF of all the three terms.

GCD stands for greatest common factor.

Here, $2y$ is GCF of $8x^{2}y-24xy+18y$.

So, the expression can be written as $2y\left(4x^2-12x+9\right)$.

Step 3. Find if it is a perfect trinomial.

A perfect trinomial is of the form $a^2\pm 2ab+b^2$.

Consider $a=2x,b=3$.

Following result can be obtained.

 $\begin{array}{rcl}2y\left(4x^2-12x+9\right)& =& 2y\left({\left(2x\right)}^2-2\left(2x\right)\left(3\right)+3^2\right)\\ & =& 2y\left(a^2-2ab+b^2\right)\end{array}$

Therefore, the expression is a perfect trinomial and can be expressed as $2y{\left(2x-3\right)}^2$.

Step 4. Verify the result.

Multiply the factors as follows.

$\begin{array}{rcl}2y{\left(2x-3\right)}^2& =& 2y\left({\left(2x\right)}^2-2\left(2x\right)\left(3\right)+3^2\right)\\ & =& 2y\left(4x^2-12x+9\right)\\ & =& 8x^{2}y-24xy+18y\end{array}$